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We denote by $c_t^{(m)}(n)$ the coefficient of $q^n$ in the series expansion of $(q;q)_infty^m(q^t;q^t)_infty^{-m}$, which is the $m$-th power of the infinite Borwein product. Let $t$ and $m$ be positive integers with $m(t-1)leq 24$. We provide asymptotic formula for $c_t^{(m)}(n)$, and give characterizations of $n$ for which $c_t^{(m)}(n)$ is positive, negative or zero. We show that $c_t^{(m)}(n)$ is ultimately periodic in sign and conjecture that this is still true for other positive integer values of $t$ and $m$. Furthermore, we confirm this conjecture in the cases $(t,m)=(2,m),(p,1),(p,3)$ for arbitrary positive integer $m$ and prime $p$.
We extend the axiomatization for detecting and quantifying sign changes of Meher and Murty to sequences of complex numbers. We further generalize this result when the sequence is comprised of the coefficients of an $L$-function. As immediate applicat
In this paper, we study properties of the coefficients appearing in the $q$-series expansion of $prod_{nge 1}[(1-q^n)/(1-q^{pn})]^delta$, the infinite Borwein product for an arbitrary prime $p$, raised to an arbitrary positive real power $delta$. We
We study sign changes in the sequence ${ A(n) : n = c^2 + d^2 }$, where $A(n)$ are the coefficients of a holomorphic cuspidal Hecke eigenform. After proving a variant of an axiomatization for detecting and quantifying sign changes introduced by Meher
For a given sequence $mathbf{alpha} = [alpha_1,alpha_2,dots,alpha_{N+1}]$ of $N+1$ positive integers, we consider the combinatorial function $E(mathbf{alpha})(t)$ that counts the nonnegative integer solutions of the equation $alpha_1x_1+alpha_2 x_2+c
In the past two decades, many researchers have studied {it index $2$} Gauss sums, where the group generated by the characteristic $p$ of the underling finite field is of index $2$ in the unit group of ${mathbb Z}/m{mathbb Z}$ for the order $m$ of the